Matrix Preconditioner Research Articles

Integration matrix based on arbitrary grids with a preconditioner for pseudospectral method

A matrix representation of integration for arbitrary grids is introduced. Suitable results are then obtained to be used along with differentiation matrix preconditioner to implement Pseudospectral method on integro-differential equations using arbitrary grids. Numerical examples are given to clarify the efficiency of the new method.

Hierarchical matrix preconditioners for the Oseen equations

Hierarchical matrices provide a technique for the data-sparse approximation and matrix arithmetic of large, fully populated matrices. In particular, approximate inverses as well as approximate LU factorizations of finite element stiffness matrices may be computed and stored in nearly optimal complexity. In this paper, we develop efficient $$\mathcal{H}$$-matrix preconditioners for the Oseen equations. In particular, $$\mathcal{H}$$-matrices will provide efficient preconditioners for the auxiliary (scalar) discrete convection–diffusion and pressure Schur complement problems. We will provide various numerical tests comparing the resulting preconditioners with each other.

Analysis of block matrix preconditioners for elliptic optimal control problems

AbstractIn this paper, we describe and analyse several block matrix iterative algorithms for solving a saddle point linear system arising from the discretization of a linear‐quadratic elliptic control problem with Neumann boundary conditions. To ensure that the problem is well posed, a regularization term with a parameter α is included. The first algorithm reduces the saddle point system to a symmetric positive definite Schur complement system for the control variable and employs conjugate gradient (CG) acceleration, however, double iteration is required (except in special cases). A preconditioner yielding a rate of convergence independent of the mesh size h is described for Ω ⊂ R2 or R3, and a preconditioner independent of h and α when Ω ⊂ R2. Next, two algorithms avoiding double iteration are described using an augmented Lagrangian formulation. One of these algorithms solves the augmented saddle point system employing MINRES acceleration, while the other solves a symmetric positive definite reformulation of the augmented saddle point system employing CG acceleration. For both algorithms, a symmetric positive definite preconditioner is described yielding a rate of convergence independent of h. In addition to the above algorithms, two heuristic algorithms are described, one a projected CG algorithm, and the other an indefinite block matrix preconditioner employing GMRES acceleration. Rigorous convergence results, however, are not known for the heuristic algorithms. Copyright © 2007 John Wiley & Sons, Ltd.

Spectral Analysis of Nonsymmetric Quasi-Toeplitz matrices with Applications to Preconditioned Multistep Formulas

The eigenvalue spectrum of a class of nonsymmetric preconditioned matrices arising in time-dependent partial differential equations is analyzed and discussed. The matrices generated by the underlying numerical integrators are small rank perturbations of block Toeplitz matrices; circulant-like preconditioners based on the former are considered. The eigenvalue distribution of the preconditioned matrix influences often crucially the convergence of Krylov iterative accelerators. Due to several reasons (lack of symmetry, band structure, and coefficients depending on the size) the classical approach based on smooth generating functions gives very little insight here. Therefore, to characterize the eigenvalues, a difference equation approach exploiting the band Toeplitz and circulant patterns generalizing the well-known results of Trench is proposed.

Block and joint ℋ︁‐matrix preconditioners for the Oseen equations

AbstractFor saddle point problems in fluid dynamics, many preconditioners in the literature exploit the block structure of the problem to construct block diagonal or block triangular preconditioners. The performance of such preconditioners depends on whether fast, approximate solvers for the linear systems on the block diagonal as well as for the Schur complement are available. We will construct these efficient preconditioners using hierarchical matrix techniques in which fully populated matrices are approximated by blockwise low rank approximations. We will compare such block preconditioners with those obtained through a completely different approach where the given block structure is not used but a domain‐decomposition based ℋ︁‐LU factorization is constructed for the complete system matrix. Preconditioners resulting from these two approaches will be discussed and compared through numerical results. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Investigation of a preconditioner for the P2 − P1 finite element solution of Stokes problem

Discretizing the classical Stokes problem in the P 2 − P 1 finite element method gives rise to a symmetric indefinite stiffness matrix. In this paper a suitable preconditioner matrix is considered to be used in the preconditioned conjugate residual and it is shown that it can produce very reasonable condition number.

Optimized linear FEM for modeling deformable objects

AbstractThe main contribution of our work is a technique to accelerate the real‐time simulation of deformable objects using a stable finite element method. Our purpose is to construct a preconditioner matrix for the coefficient matrix of the implicit integration scheme based on the global rotation matrix of the object. We describe some required steps to compute an optimal preconditioner in an off‐line step. Our technique can be used to accelerate the element stiffness warping method and could be used to solve the system directly for objects with small deformations or for objects that appear outside of the users' interest zone. Copyright © 2006 John Wiley & Sons, Ltd.

A preconditioner for the Schur complement matrix

A preconditioner for iterative solution of the interface problem in Schur Complement Domain Decomposition Methods is presented. This preconditioner is based on solving a global problem in a narrow strip around the interface. It requires much less memory and computing time than classical Neumann–Neumann preconditioner and its variants, and handles correctly the flux splitting among subdomains that share the interface. The aim of this work is to present a theoretical basis (regarding the behavior of Schur complement matrix spectra) and some simple numerical experiments conducted in a sequential environment as a motivation for adopting the proposed preconditioner. Efficiency, scalability, and implementation details on a production parallel finite element code [Sonzogni V, Yommi A, Nigro N, Storti M. A parallel finite element program on a Beowulf cluster. Adv Eng Software 2002;33(7–10):427–43; Storti M, Nigro N, Paz R, Dalcín L. PETSc-FEM: a general purpose, parallel, multi-physics FEM program, 1999–2006] can be found in works [Paz R, Storti M. An interface strip preconditioner for domain decomposition methods: application to hydrology. Int J Numer Methods Eng 2005;62(13):1873–94; Paz R, Nigro N, Storti M. On the efficiency and quality of numerical solutions in cfd problems using the interface strip preconditioner for domain decomposition methods. Int J Numer Methods Fluids, in press].

A Comparison of Data Assimilation Methods Using a Planetary Geostrophic Model

Abstract Assimilating hydrographic observations into a planetary geostrophic model is posed as a problem in control theory. The cost functional is the sum of weighted model and data residuals. Model errors are assumed to be spatially correlated, and hydrographic station data are assimilated directly. Searches in state space and data space, for minimizing the cost functional, are compared to a direct matrix inversion algorithm in the data space. State-space methods seek the minimizer of the cost functional by performing a preconditioned search in an N-dimensional space of state or control variables, where N is approximately 650 000 in the present calculations. Data-space methods solve the Euler–Lagrange equations for the extremum of the cost functional by working in an M-dimensional dual space, where M is the number of measurements. The following four solvers are compared: (i) an iterative state-space solver, with a naive diagonal matrix preconditioner; (ii) an iterative state-space solver, with a sophisticated preconditioner based on the inverse of the model’s dynamical operators; (iii) an iterative data-space solver, with no preconditioning; and (iv) a direct, M × M matrix inversion, data-space solver. The best solver is the iterative data-space solver, (iii), which is approximately 10 times faster than the sophisticated preconditioned state-space solver, (ii), and 100 times faster than the direct data-space solver, (iv).

Iterative time independent calculation of the cumulative reaction probability within a basis adapted preconditioner

A band preconditioner matrix coupled to an iterative approach based on the generalized minimal residual (GMRes) method is presented to determine the cumulative reaction probability (CRP) N( E). The CRP is calculated using the Seideman, Manthe and Miller Lanczos-based boundary condition method [J. Chem. Phys. 96 (1992) 4412; 99 (1993) 3411]. Using this basis adapted preconditioner, the iterative GMRes scheme is found to be more efficient than a direct method based on the LU decomposition. The efficiency of this approach is illustrated by calculating the CRP for the H + O 2 → HO + O reaction, assuming zero total angular momentum.

Symmetric Gauss-Seidel multigrid solution of the Euler equations on structured and unstructured grids

The efficient symmetric Gauss-Seidel (SGS) algorithm for solving the Euler equations of inviscid, compressible flow on structured grids, developed in collaboration with Jameson of Stanford University, is extended to unstructured grids. The algorithm uses a nonlinear formulation of an SGS solver, implemented within the framework of multigrid. The earlier form of the algorithm used the natural (lexicographic) ordering of the mesh cells available on structured grids for the SGS sweeps, but a number of features of the method that are believed to contribute to its success can also be implemented for computations on unstructured grids. The present paper reviews, the features of the SGS multigrid solver for structured gr0ids, including its nonlinear implementation, its use of “absolute” Jacobian matrix preconditioning, and its incorporation of multigrid, and then describes the incorporation of these features into an algorithm suitable for computations on unstructured grids. The implementation on unstructured grids is based on the agglomerated multigrid method developed by Sørensen, which uses an explicit Runge-Kutta smoothing algorithm. Results of computations for steady, transonic flows past two-dimensional airfoils are presented, and the efficiency of the method is evaluated for computations on both structured and unstructured meshes.

Multi-frequency diffuse optical tomography

The performance of diffuse optical tomography (DOT) is investigated when multi-modulation frequencies are used simultaneously in the inverse problem in order to cast the forward model. The forward model used was based on the first-order Rytov approximation of the diffusion equation. Numerical measurements were generated with a finite difference approach for configurations relevant to breast optical mammography. The impact of various frequency-set characteristics was evaluated. Furthermore, the effect of sensitivity matrix preconditioning was investigated. The criteria underlying this study were based on a ‘singular value analysis’ and parameter cross-talk between absorption and scattering reconstructions. It had been shown that preconditioning the matrix was necessary to provide accurate results and an even distribution of frequencies within the larger bound. Furthermore, correlating the different criterions used demonstrated that analysis based on optimization of condition number alone was providing misleading results.

Static reanalysis of structures with added degrees of freedom

AbstractThis paper deals with static reanalysis of a structure with added degrees of freedom where the nodes of the original structure form a subset of the nodes of the modified structure. A preconditioned conjugate‐gradient approach is developed. The preconditioner is constructed, and the implementation of the approach involves only decomposition of the stiffness matrix corresponding to the newly added degrees of freedom. In particular, the approach can adaptively monitor the accuracy of approximate solutions. The approach is applicable to the reanalysis of the structural layout modifications for the case of addition of some nodes, deletion and addition of elements and further changes in the geometry as well as to the local mesh refinements. Numerical examples show that the condition number of the selected preconditioned matrix is largely reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2005 John Wiley & Sons, Ltd.

Optimized interface conditions in domain decomposition methods for problems with extreme contrasts in the coefficients

When the coefficients of a problem have jumps of several orders of magnitude and are anisotropic, many preconditioners and domain decomposition methods (DDM) suffer from plateaus in the convergence due to the presence of very small isolated eigenvalues in the spectrum of the preconditioned linear system. One way to improve the preconditioner is to use a linear algebra technique called deflation, or very similarly coarse grid corrections. In both cases, it is necessary to identify and compute, at least approximately, all the eigenvectors corresponding to the “bad” eigenvalues. In the framework of DDM, we propose a way to design interface conditions so that convergence is fast and does not have any plateau. The method relies only on the knowledge of the smallest and largest eigenvalues of an auxiliary matrix. The eigenvectors are not used. The method relies on van der Sluis’ result on a quasi-optimal diagonal preconditioner for a symmetric positive definite matrix. It is then possible to design Robin interface conditions using only one real parameter for the entire interface. By adding a second real parameter and more general interface conditions, it is possible to take into account highly heterogeneous and anisotropic media. Numerical results are given and compared with other approaches.

Matrix conditioning for lengthened capillary DNA sequencing

Capillary electrophoresis (CE) is currently the preferred format for both DNA sequencing and small DNA fragment analysis. The present study provides a simple revision of the procedure used for CE of DNA with a commercial DNA sequencing apparatus from Applied Biosystems. The revision is electrophoretic conditioning of the sieving matrix (typically POP-6) before sample injection. The effects of this preconditioning are revealed during subsequent analyses performed without replenishing the sieving matrix. The primary effect of preconditioning is to increase peak separations during a subsequent CE. The preconditioning has the following characteristics: (i) The effect on peak separation progressively increases as the preconditioning time increases to at least 6 h. (ii) The effect on peak separation scales approximately as the product of the preconditioning time and the magnitude of the electrical field (162 - 320 V/cm) during preconditioning. (iii) The preconditioning persists for more than 72 h at zero field. Preconditioning of the matrix substantially improves resolution of fragment analysis in the range of 700-2000 nucleotides. For DNA sequencing, the primary impact of preconditioning is, thus far, extension of the range of low-quality base calls at the end of sequence reading. Matrix preconditioning is a new factor to consider when interpreting data obtained by CE in polymer solutions. The mechanism of preconditioning is not yet known.

On the Non‐Hierarchical Preconditioner Constructions

AbstractIn this paper a new cyclic matrix representation of the H–1/2 norm is presented. Its application as Schur complement preconditioning matrix requires only matrix‐vector multiplication. The computational cost of this matrix‐vector multiplication is O (N · log(N)) arithmetic operations, where N is the number of unknowns. The efficiency of the construction to elliptic problems has been verified by numerical tests. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A new preconditioned conjugate gradient power flow

A new solution methodology for the constant matrix, decoupled power flow problem is presented in this paper. The proposed method uses the conjugate gradient method instead of the traditional direct solution of Ax=b for updating the power flow variables. The conjugate gradient method is accelerated with an approximate inverse matrix preconditioner obtained from a linear combination of matrix-valued Chebyshev polynomials. The new method has been tested on several systems of different sizes. In terms of speed, the method is comparable to the fast decoupled load flow in serial environments but it is more amenable to parallel and vector processing since it contains only matrix-vector multiplications.

Rational approximation preconditioners for sparse linear systems

This paper presents a class of preconditioning techniques which exploit rational function approximations to the inverse of the original matrix. The matrix is first shifted and then an incomplete LU factorization of the resulting matrix is computed. The resulting factors are then used to compute a better preconditioner for the original matrix. Since the incomplete factorization is made on a shifted matrix, a good LU factorization is obtained without allowing much fill-in. The result needs to be extrapolated to the nonshifted matrix. Thus, the main motivation for this process is to save memory. The method is useful for matrices whose incomplete LU factorizations are poor, e.g., unstable.

Analysis of Millimeter Wave Scattering by the Infinite Plane Metallic Grating Using PCG-FFT Technique

In this paper, both fast Fourier transformation (FFT) and preconditioned CG technique are introduced into method of lines (MOL) to further enhance the computational efficiency of this semi-analytic method. Electromagnetic wave scattering by an infinite plane metallic grating is used as the examples to describe its implementation. For arbitrary incident wave, Helmholz equation and boundary condition are first transformed into new ones so that the impedance matrix elements are calculated by FFT technique. As a result, this Topelitz impedance matrix only requires O(N) memory storage for the conjugate gradient FFT method to solve the current distribution with the computational complexity O(N log N) . Our numerical results show that circulate matrix preconditioner can speed up CG-FFT method to converge in much smaller CPU time than the banded matrix preconditioner.

Multilevel diagonal scaling preconditioners for boundary element equations on locally refined meshes

We study a multilevel preconditioner for the Galerkin boundary element matrix arising from a symmetric positive-definite bilinear form. The associated energy norm is assumed to be equivalent to a Sobolev norm of positive, possibly fractional, order m on a bounded (open or closed) surface of dimension d, with \(0<2m\le d\). We consider piecewise linear approximation on triangular elements. Successive levels of the mesh are created by selectively subdividing elements within local refinement zones. Hanging nodes may be created and the global mesh ratio can grow exponentially with the number of levels. The coarse-grid correction consists of an exact solve, and the correction on each finer grid amounts to a simple diagonal scaling involving only those degrees of freedom whose associated nodal basis functions overlap the refinement zone. Under appropriate assumptions on the choice of refinement zones, the condition number of the preconditioned system is shown to be bounded by a constant independent of the number of degrees of freedom, the number of levels and the global mesh ratio. In addition to applying to Galerkin discretisation of hypersingular boundary integral equations, the theory covers finite element methods for positive-definite, self-adjoint elliptic problems with Dirichlet boundary conditions.